Made glorious summer (by the sons of Yvettes*)

And all the clouds that lour'd upon our sky
(this post has to be read as a continuation of the one from yesterday)
Noncommutative geometry was shown to provide a promising framework for unification of all fundamental interactions including gravity [3,5,6,10,12]. Historically, the search to identify the structure of the noncommutative space followed the bottom-up approach where the known spectrum of the fermionic particles was used to determine the geometric data that defines the space. This bottom-up approach involved an interesting interplay with experiments. While at first the experimental evidence of neutrino oscillations contradicted the first attempt [6], it was realized several years later in 2006 ([12]) that the obstruction to get neutrino oscillations was naturally eliminated by dropping the equality between the metric dimension of space-time (which is equal to 4 as far as we know) and its KO-dimension which is only defined modulo 8. When the latter is set equal to 2 modulo 8 [2,4] (using the freedom to adjust the geometry of the finite space encoding the fine structure of space-time) everything works fine, the neutrino oscillations are there as well as the see-saw mechanism which appears for free as an unexpected bonus. Incidentally, this also solved the fermionic doubling problem by allowing a simultaneous Weyl-Majorana condition on the fermions to halve the degrees of freedom. 
The second interplay with experiments occurred a bit later when it became clear that the mass of the Brout-Englert-Higgs boson would not comply with the restriction (that mH 170 Gev) imposed by the validity of the Standard Model up to the unification scale. This obstruction to lower mH was overcome in [11] simply by taking into account a scalar field which was already present in the full model which we had computed previously in [10]. One lesson which we learned on that occasion is that we have to take all the fields of the noncommutative spectral model seriously, without making assumptions not backed up by valid analysis, especially because of the almost uniqueness of the Standard Model (SM) in the noncommutative setting.
The SM continues to conform to all experimental data. The question remains whether this model will continue to hold at much higher energies, or whether there is a unified theory whose low-energy limit is the SM. One indication that there must be a new higher scale that effects the low energy sector is the small mass of the neutrinos which is explained through the see-saw mechanism with a Majorana mass of at least of the order of 1011Gev. In addition and as noted above, a scalar field which acquires a vev generating that mass scale can stabilize the Higgs coupling and prevent it from becoming negative at higher energies and thus make it consistent with the low Higgs mass of 126 Gev [11]. Another indication of the need to modify the SM at high energies is the failure (by few percent) of the three gauge couplings to be unified at some high scale which indicates that it may be necessary to add other matter couplings to change the slopes of the running of the RG equations.
This leads us to address the issue of the breaking from the natural algebra A which results from the classification of irreducible finite geometries of KO-dimension 6 (modulo 8) performed in [9], to the algebra corresponding to the SM. This breaking was effected in [8, 9] using the requirement of the first order condition on the Dirac operator. The first order condition is the requirement that the Dirac operator is a derivation of the algebra A into the commutant of Â=JAJ-1 where J is the charge conjugation operator. This in turn guarantees the gauge invariance and linearity of the inner fluctuations [7] under the action of the gauge group given by the unitaries U=uJuJ-1 for any unitary u∈A. This condition was used as a mathematical requirement to select the maximal subalgebra M3()RLM4() which is compatible with the first order condition and is the main reason behind the unique selection of the SM. The existence of examples of noncommutative spaces where the first order condition is not satisfied such as quantum groups and quantum spheres provides a motive to remove this condition from the classification of noncommutative spaces compatible with unification [14,15,16,17]. This study was undertaken in a companion paper [13] where it was shown that in the general case the inner fluctuations of D form a semigroup in the product algebra AAop, and acquire a quadratic part in addition to the linear part. Physically, this new phenomena will have an impact on the structure of the Higgs fields which are the components of the connection along discrete directions. This paper is devoted to the construction of the physical model that describes the physics beyond the Standard Model. The methods used build on previous results and derivations developed over the years.
(Submitted on 30 Apr 2013 (v1), last revised 25 Sep 2014 (this version, v4))

In the finite bosom of duration unfold.
In particle physics there is a well accepted notion of particle which is the same as that of irreducible representation of the Poincaré group. It is thus natural to expect that the notion of particle in Quantum Gravity will involve irreducible representations in Hilbert space, and the question is "of what?". 

What we have found is a candidate answer which is a degree 4 analogue of the Heisenberg canonical commutation relation [p,q]=i. The degree 4 is related to the dimension of space-time. The role of the operator p is now played by the Dirac operator D. The role of q is played by the Feynman slash of real fields, so that one applies the same recipe to spatial variables as one does to momentum variables. The equation is then of the form E(Z[D,Z]4)=γ where γ is the chirality and where the E of an operator is its projection on the commutant of the gamma matrices used to define the Feynman slash.

Our main results then are that:

1) Every spin 4-manifold M (smooth compact connected) appears as an irreducible representation of our two-sided equation.

2) The algebra generated by the slashed fields is the algebra of functions on M with values in A=M2(ℍ)M4(ℂ), which is exactly the slightly noncommutative algebra needed to produce gravity coupled to the Standard Model minimally extended to an asymptotically free theory

3) The only constraint on the Riemannian metric of the 4-manifold is that its volume 
is quantized, which means that it is an integer (larger than 4) in Planck units.
The great advantage of 3) is that, since the volume is quantized, the huge cosmological term which dominates the spectral action is now quantized and no longer interferes with the equations of motion which as a result of our many years collaboration with Ali Chamseddine gives back the Einstein equations coupled with the Standard Model.

The big plus of 2) is that we finally understand the meaning of the strange choice of algebras that seems to be privileged by nature: it is the simplest way of replacing a number of coordinates by a single operator. Moreover as the result of our collaboration with Walter van Suijlekom, we found that the slight extension of the SM to a Pati-Salam model given by the algebra M2(ℍ)M4(ℂ) greatly improves things from the 
mathematical standpoint while moreover making the model asymptotically free!
To get a mental picture of the meaning of 1), I will try an image which came gradually while we were working on the problem of realizing all spin 4-manifolds with arbitrarily large quantized volume as a solution to the equation. 

"The Euclidean space-time history unfolds to macroscopic dimension from the product of two 4-spheres of Planckian volume as a butterfly unfolds from its chrysalis."
Posted by Alain Connes at 4:36 PM

* I hope the reader will forgive this clumsy hijacking of the words of Shakespeare and the private joke about the river flowing near the Institut des Hautes Etudes Scientifiques where Chamseddine and Connes work.


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